FIGURE 6.7. Lord’s paradox. (Ellipses represent scatter plots of data.) As a whole, neither boys nor girls gain weight during the year, but in each stratum of the initial weight, boys tend to gain more than girls.

The second statistician, on the other hand, argues that because the final weight of a student is strongly influenced by his or her initial weight, we should stratify the students by initial weight. If you make a vertical slice through both ellipses, which corresponds to looking only at the boys and girls with a particular value of the initial weight (say W0 in Figure 6.7), you will notice that the vertical line intersects the Boys ellipse higher up than it does the Girls ellipse, although there is a certain amount of overlap. This means that boys who started with weight W0 will have, on average, a higher final weight (WF) than the girls who started with weight W0. Accordingly, Lord writes, “the second statistician concludes, as is customary in such cases, that the boys showed significantly more gain in weight than the girls when proper allowance is made for differences in initial weight between the sexes.”

What is the school’s dietitian to do? Lord writes, “The conclusions of each statistician are visibly correct.” That is, you don’t have to crunch any numbers to see that two solid arguments are leading to two different conclusions. You need only look at the figure. In Figure 6.7, we can see that boys gain more weight than girls in every stratum (every vertical cross section). Yet it’s equally obvious that both boys and girls gained nothing overall. How can that be? Is not the overall gain just an average of the stratum-specific gains?

Now that we are experienced pros at the fine points of Simpson’s paradox and the sure-thing principle, we know what is wrong with that argument. The sure-thing principle works only in cases where the relative proportion of each subpopulation (each weight class) does not change from group to group. Yet, in Lord’s case, the “treatment” (gender) very strongly affects the percentage of students in each weight class.

So we can’t rely on the sure-thing principle, and that brings us back to square one. Who is right? Is there or isn’t there a difference in the average weight gains between boys and girls when proper allowance is made for differences in the initial weight between the sexes? Lord’s conclusion is very pessimistic: “The usual research study of this type is attempting to answer a question that simply cannot be answered in any rigorous way on the basis of available data.” Lord’s pessimism spread beyond statistics and has led to a rich and quite pessimistic literature in epidemiology and biostatistics on how to compare groups that differ in “baseline” statistics.

I will show now why Lord’s pessimism is unjustified. The dietitian’s question can be answered in a rigorous way, and as usual the starting point is to draw a causal diagram, as in Figure 6.8. In this diagram, we see that Sex (S) is a cause of initial weight (WI) and final weight (WF).

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